3. Implemented in Code_Aster#

3.1. Practical implementation#

The operator CALC_ERC_DYN [U4.53.41] was developed to solve the problem of finding optimal fields \((u,v,w)\) associated with the constrained optimization problem described in the previous section.

Concretely, for a list of given frequencies, it solves the system of linear equations \(Al=b\) detailed above, which makes it possible to obtain the fields \(u\) and \(u-v\) sought. It should be noted that field \(u-w\) is not proposed as the output of the operator because it is trivially the result of a proportional relationship with \(u-v\).

This operator also makes it possible to evaluate, if desired, the value of the functional \({e}_{\omega }^{2}\) in the \((u,v,w)\text{}\) solution triple. This is particularly interesting when you want to use the \({e}_{\omega }^{2}\) function in a realignment or identification problem.

The operator uses five mandatory keywords:

  • the keywords MATR_RIGI and MATR_MASSE correspond respectively to the real assembled stiffness and mass matrices of the model studied;

  • the keyword MATR_NORME: this keyword allows you to fill in the generalized assembled matrix corresponding to the positive definite symmetric matrix \({G}_{r}\) serving as the norm of errors in the observation space.

  • the keyword MATR_PROJ: this keyword is used to give the projection matrix associated with the \(H\) operator. It is a geometric correspondence between the mesh associated with the mass and stiffness matrices on the one hand, and the mesh associated with the \(\widehat{u}\) observations on the other hand. The projection matrix requested here must be the result of a calculation with the operator PROJ_CHAMP [U4.72.05] with the method “COLLOCATION”.

  • the MESURE keyword: this keyword allows you to fill in the fields that will be used as an \(\widehat{{u}_{i}}\) observation. This concept should contain as many order numbers as the number of frequencies that will be studied when the operator is called (operands FREQ/LIST_FREQ). Moreover, the dimension of the fields contained in this concept must be consistent with the dimension of the standard matrix entered under the operand MATR_NORME. In addition, the fields contained must not include Lagrange-type degrees of freedom. To do this, it is advisable to condition this concept using the operator OBSERVATION [U4.90.03] beforehand;

  • the CHAMP_MESURE keyword to define the type of field contained in the measurement;

  • the FREQ/LIST_FREQ keyword to define the frequencies at which the problem will be solved;

  • the keywords ALPHAet GAMMA corresponding to the weighting parameters of the functional comparable to a regularization term (\(\alpha\)) and to the error weighting parameter \((u-v)\) and \((u-w)\) (\(\gamma\));

  • the EVAL_FONCT keyword to determine if the functional value for the optimal field triplet will be evaluated and stored in the result;

  • the SOLVEURpermettant keyword to choose the type of solver that will be used when solving the problem. In the current release, the solvers available are MUMPS (default) and LDLT.